Mathematicians Prove Symmetry of Phase Transitions-II (Quanta)

Subhotosh Khan

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continued from - Mathematicians Prove Symmetry of Phase Transitions-I (Quanta)
The First Proofs
In 2001 Smirnov produced the first rigorous mathematical proof of conformal invariance in a physical model. It applied to a model of percolation, which is the process of liquid passing through a maze in a porous medium, like a stone.

Smirnov looked at percolation on a triangular lattice, where water is allowed to flow only through vertices that are “open.” Initially, every vertex has the same probability of being open to the flow of water. When the probability is low, the chances of water having a path all the way through the stone is low.

But as you slowly increase the probability, there comes a point where enough vertices are open to create the first path spanning the stone. Smirnov proved that at the critical threshold, the triangular lattice is conformally invariant, meaning percolation occurs regardless of how you transform it with conformal symmetries.

Since it used magic, it only works in situations where there is magic, and we weren’t able to find magic in other situations.

Five years later, at the 2006 International Congress of Mathematicians, Smirnov announced that he had proved conformal invariance again, this time in the Ising model. Combined with his 2001 proof, this groundbreaking work earned him the Fields Medal, math’s highest honor.

In the years since, other proofs have trickled in on a case-by-case basis, establishing conformal invariance for specific models. None have come close to proving the universality that Polyakov envisioned.

“The previous proofs that worked were tailored to specific models,” said Federico Camia, a mathematical physicist at New York University Abu Dhabi. “You have a very specific tool to prove it for a very specific model.”

Smirnov himself acknowledged that both of his proofs relied on some sort of “magic” that was present in the two models he worked with but isn’t usually available.

“Since it used magic, it only works in situations where there is magic, and we weren’t able to find magic in other situations,” he said.

The new work is the first to disrupt this pattern — proving that rotational invariance, a core feature of conformal invariance, exists widely.

One at a Time
Duminil-Copin first began to think about proving universal conformal invariance in the late 2000s, when he was Smirnov’s graduate student at the University of Geneva. He had a unique understanding of the brilliance of his mentor’s techniques — and also of their limitations. Smirnov bypassed the need to prove all three symmetries separately and instead found a direct route to establishing conformal invariance — like a shortcut to a summit.

“He’s an amazing problem solver. He proved conformal invariance of two models of statistical physics by finding the entrance in this huge mountain, like this kind of crux that he went through,” said Duminil-Copin.

For years after graduate school, Duminil-Copin worked on building up a set of proofs that might eventually allow him to go beyond Smirnov’s work. By the time he and his co-authors set to work in earnest on conformal invariance, they were ready to take a different approach than Smirnov had. Rather than take their chances with magic, they returned to the original hypotheses about conformal invariance made by Polyakov and later physicists.


The physicists had required a proof in three steps, one for each symmetry present in conformal invariance: translational, rotational and scale invariance. Prove each of them separately, and you get conformal invariance as a consequence.

With this in mind, the authors set out to prove scale invariance first, believing that rotational invariance would be the most difficult symmetry and knowing that translational invariance was simple enough and wouldn’t require its own proof. In attempting this, they realized instead that they could prove the existence of rotational invariance at the critical point in a large variety of percolation models on square and rectangular grids.

They used a technique from probability theory, called coupling, that made it possible to directly compare the large-scale behavior of square lattices with rotated rectangular lattices. By combining this approach with ideas from another field of mathematics called integrability, which studies hidden structures in evolving systems, they were able to prove that the behavior at critical points was the same across the models — thus establishing rotational invariance. Then they proved that their results extended to other physical models where it’s possible to apply the same coupling.

The end result is a powerful proof that rotational invariance is a universal property of a large subset of known two-dimensional models. They believe the success of their work indicates that a similarly eclectic set of techniques, melded from various fields of math, will be necessary to make additional progress on conformal invariance.

“I think it’s going to be more and more true, in arguments of conformal invariance and the study of phase transitions, that you need a little bit of everything. You cannot just attack it with one angle of attack,” said Duminil-Copin.

Last Steps
For the first time since Smirnov’s 2001 result, mathematicians have new purchase on the long-standing challenge of proving the universality of conformal invariance. And unlike that earlier work, this new result opens new paths to follow. By following a bottom-up approach in which they aimed to prove one constituent symmetry at a time, the researchers hope they laid a foundation that will eventually support a universal set of results.

Now, with rotational invariance down, Duminil-Copin and his colleagues have their sights set on scale invariance, their original target. A proof of scale invariance, given the recent work on rotational symmetry and the fact that translational symmetry doesn’t need its own proof, would put mathematicians on the cusp of proving full conformal invariance. And the flexibility of their methods makes the researchers optimistic it can be done.

“I definitely think that step three is going to fall fairly soon,” said Duminil-Copin. “If it’s not us, it would be somebody smarter, but definitely, it’s going to happen very soon.”

The proof of rotational invariance took five years, though, so the next results may yet take some time. Still, Smirnov is hopeful that two-dimensional conformal invariance may finally be within reach.

“That might mean a week, or it might mean five years, but I’m much more optimistic than I was in November,” said Smirnov.

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